If type
is "stationary"
, then the stationary
bootstrap scheme with mean block length b
according to Politis
and Romano (1994) is computed. For type
equals "block"
,
the blockwise bootstrap with block length b
according to
Kuensch (1989) is used.
If m > 1
, then the block of blocks bootstrap is computed
(see Kuensch, 1989). The basic sampling scheme is the same as for
the case m = 1
, except that the bootstrap is applied to a series
y
containing blocks of length m
, where each block of y
is
defined as \(y[t] = (x[t], \dots, x[t-m+1])\). Therefore, for the block
of blocks bootstrap the first argument of statistic
is given by
a n x m
matrix yb
, where each row of yb
contains one
bootstrapped basic block observation \(y[t]\) (n
is the number of
observations in x
).
Note, that for statistics which are functions of the empirical
m
-dimensional marginal (m > 1
) only this procedure
yields asymptotically valid bootstrap estimates. The
case m = 1
may only be used for symmetric statistics (i.e., for
statistics which are invariant under permutations of x
).
tsboot
does not implement the block of blocks
bootstrap, and, therefore, the first example in tsboot
yields inconsistent estimates.
For consistency, the (mean) block length b
should grow with
n
at an appropriate rate. If b
is not given, then a
default growth rate of const * n^(1/3)
is used. This rate is
"optimal" under certain conditions (see the references for more
details). However, in general the growth rate depends on the specific
properties of the data generation process. A default value for
const
has been determined by a Monte Carlo simulation using a
Gaussian AR(1) process (AR(1)-parameter of 0.5, 500
observations). const
has been chosen such that the mean square
error for the bootstrap estimate of the variance of the empirical mean
is minimized.
Note, that the computationally intensive parts are fully implemented
in C
which makes tsbootstrap
about 10 to 30 times faster
than tsboot
.
Missing values are not allowed.
There is a special print method for objects of class
"resample.statistic"
which by default uses
max(3, getOption("digits") - 3)
digits to format real numbers.